Copyright © 2012 Pearson Education Inc. Gravitation Physics 7C lecture 17 Tuesday December 3, 8:00 AM – 9:20 AM Engineering Hall 1200
Copyright © 2012 Pearson Education Inc. Introduction What can we say about the motion of the particles that make up Saturn’s rings? Why doesn’t the moon fall to earth, or the earth into the sun? By studying gravitation and celestial mechanics, we will be able to answer these and other questions.
Copyright © 2012 Pearson Education Inc. Newton’s law of gravitation Law of gravitation: Every particle of matter attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
Copyright © 2012 Pearson Education Inc. Newton’s law of gravitation The gravitational force can be expressed mathematically as F g = Gm 1 m 2 /r 2, where G is the gravitational constant. Note G is different from g. G = 6.67 E -11 N m 2 /kg 2
Copyright © 2012 Pearson Education Inc. Gravitation and spherically symmetric bodies The gravitational interaction of bodies having spherically symmetric mass distributions is the same as if all their mass were concentrated at their centers. This is exact, not approximation! Let’s prove it mathematically.
Copyright © 2012 Pearson Education Inc. Gravitation and spherically symmetric bodies
Copyright © 2012 Pearson Education Inc. Gravitation and spherically symmetric bodies
Copyright © 2012 Pearson Education Inc. Gravitation and spherically symmetric bodies
Copyright © 2012 Pearson Education Inc. Determining the value of G In 1798 Henry Cavendish made the first measurement of the value of G. Figure below illustrates his method. G = 6.67 E -11 N m 2 /kg 2
Copyright © 2012 Pearson Education Inc. Some gravitational calculations F g = Gm 1 m 2 /r 2,
Copyright © 2012 Pearson Education Inc. The sphere on the right has twice the mass and twice the radius of the sphere on the left. Compared to the sphere on the left, the larger sphere on the right has A. twice the density. B. the same density. C. 1/2 the density. D. 1/4 the density. E. 1/8 the density. Q13.1 mass m radius R mass 2m radius 2R
Copyright © 2012 Pearson Education Inc. The sphere on the right has twice the mass and twice the radius of the sphere on the left. Compared to the sphere on the left, the larger sphere on the right has A. twice the density. B. the same density. C. 1/2 the density. D. 1/4 the density. E. 1/8 the density. A13.1 mass m radius R mass 2m radius 2R
Copyright © 2012 Pearson Education Inc. Why gravity is important? It is the dominant force on astronomical scale.
Copyright © 2012 Pearson Education Inc. Why gravity is important? Anomalous gravity at micron scale can be evidence for extra dimension of our universe.
Copyright © 2012 Pearson Education Inc. Weight The weight of a body is the total gravitational force exerted on it by all other bodies in the universe. At the surface of the earth, we can neglect all other gravitational forces, so a body’s weight is w = Gm E m/R E 2. The acceleration due to gravity at the earth’s surface is g = Gm E /R E 2.
Copyright © 2012 Pearson Education Inc. Weight The weight of a body decreases with its distance from the earth’s center, as shown in Figure below.
Copyright © 2012 Pearson Education Inc. Weight We can use weight to measure earth’s mass! g = Gm E /R E 2 We know the radius of earth = 6300 km from satellite/astronomical observations. Thus m E = G/g R E 2 = 5.98 E 24 kg The average density of earth is then 5500 kg/m 3, 5.5 times heavier than water!
Copyright © 2012 Pearson Education Inc. Interior of the earth The earth is approximately spherically symmetric, but it is not uniform throughout its volume. The inner core is supposed to be made of iron and rotates at high speed, giving earth’s magnetic field.
Copyright © 2012 Pearson Education Inc. Gravitational potential energy The gravitational potential energy of a system consisting of a particle of mass m and the earth is U = –Gm E m/r. This assumes zero energy at infinite distance.
Copyright © 2012 Pearson Education Inc. Gravitational potential energy U = –Gm E m/r. Proof:
Copyright © 2012 Pearson Education Inc. Gravitational potential energy depends on distance The gravitational potential energy of the earth-astronaut system increases (becomes less negative) as the astronaut moves away from the earth.
Copyright © 2012 Pearson Education Inc. From the earth to the moon To escape from the earth, an object must have the escape speed.
Copyright © 2012 Pearson Education Inc. The motion of satellites The trajectory of a projectile fired from A toward B depends on its initial speed. If it is fired fast enough, it goes into a closed elliptical orbit (trajectories 3, 4, and 5).
Copyright © 2012 Pearson Education Inc. Circular satellite orbits For a circular orbit, the speed of a satellite is just right to keep its distance from the center of the earth constant. (See Figure below.) A satellite is constantly falling around the earth. Astronauts inside the satellite in orbit are in a state of apparent weightlessness because they are falling with the satellite. (See Figure below.)